Average Error: 0.0 → 0.0
Time: 2.6s
Precision: 64
\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
↓
\[\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)\]
0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)
↓
\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)
double f(double x) {
        double r98483 = 0.70711;
        double r98484 = 2.30753;
        double r98485 = x;
        double r98486 = 0.27061;
        double r98487 = r98485 * r98486;
        double r98488 = r98484 + r98487;
        double r98489 = 1.0;
        double r98490 = 0.99229;
        double r98491 = 0.04481;
        double r98492 = r98485 * r98491;
        double r98493 = r98490 + r98492;
        double r98494 = r98485 * r98493;
        double r98495 = r98489 + r98494;
        double r98496 = r98488 / r98495;
        double r98497 = r98496 - r98485;
        double r98498 = r98483 * r98497;
        return r98498;
}
↓
double f(double x) {
        double r98499 = x;
        double r98500 = -r98499;
        double r98501 = 0.70711;
        double r98502 = 0.04481;
        double r98503 = 0.99229;
        double r98504 = fma(r98502, r98499, r98503);
        double r98505 = 1.0;
        double r98506 = fma(r98499, r98504, r98505);
        double r98507 = 0.27061;
        double r98508 = 2.30753;
        double r98509 = fma(r98507, r98499, r98508);
        double r98510 = r98506 / r98509;
        double r98511 = r98501 / r98510;
        double r98512 = fma(r98500, r98501, r98511);
        return r98512;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016 \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}\right)}\]
  3. Using strategy rm

  4. Applied associate-/l*0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \color{blue}{\frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(-x, 0.707110000000000016, \frac{0.707110000000000016}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}{\mathsf{fma}\left(0.27061000000000002, x, 2.30753\right)}}\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1 (* x (+ 0.99229 (* x 0.04481))))) x)))